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A point-in-polygon method based on a quasi-closest
point
Sheng Yang, Jun-Hai Yong, Jiaguang Sun, He-Jin Gu, Jean-Claude Paul
To cite this version:
Sheng Yang, Jun-Hai Yong, Jiaguang Sun, He-Jin Gu, Jean-Claude Paul.
A point-in-polygon
method based on a quasi-closest point. Computers & Geosciences, Elsevier, 2010, 36 (2),
pp.205-213. �10.1016/j.cageo.2009.06.008�. �inria-00517936�
ARTICLE IN PRESS
presented an efficient algorithm for the point-in-polygon pro-blem. In some cases, however, the algorithm can give incorrect answers (seeFig. 1andTable 1). The purpose of this paper is to present the criteria for the point-in-polygon tests against possibly concave polygons, and moreover, to resolve the singularity of a test point on an edge within a user-defined tolerance.
2. Related work and our improvements
To efficiently obtain the inclusion property, the algorithm named CBCA in (Zalik and Kolingerova, 2001) preprocesses a polygon by subdividing its bounding rectangle into uniform-sized cells. In addition, the preprocessing steps include recording each edge of the polygon into the cells crossed by the edge and classifying the empty cells as inside or outside the polygon. Let p be a test point and Cell(p) be the cell containing p. If Cell(p) is empty, the inclusion property of p is obtained directly from that of Cell(p); otherwise CBCA employs the orientation method to determine the inclusion property, this is, it finds the closest edge (or vertex) of p to the edges stored in Cell(p), and then determines
the inclusion property from the orientation of p with respect to the closest edge (or the convexity property of the closest vertex). As shown inFig. 1andTable 1, CBCA may produce incorrect result when the closest point is out of Cell(p) or the adjacent edges of the vertex are nearly collinear.
InFig. 1a, q2is not the closest point of p to all the edges, and more
importantly, the line segment pq2 can intersect v5v6 so that the
orientation of p with respect to v2v3 is unrelated to the inclusion
property of p. InFig. 1b, q4is not the closest point of p to the edges
crossing Cell(v4), so the convexity property of v4is unrelated to the
inclusion property of p as well. InFig. 1c, v5is the closest point in the
direction from p to v5, but it is difficult to determine the convexity
property of v5in the presence of rounding errors; especially, v5is not
the closest point of p to all the edges because at least the distance between p and v2is smaller than that between p and v5. It can be seen
that finding the closest point in any direction is more expensive than in a certain direction. Underlying these observations are the needs for finding such a parameter that it is not only suitable for the orientation method but also easy to be evaluated through the neighboring cells of a test point, and avoiding the use of the convexity property of a vertex when its adjacent edges are nearly collinear.
v1 = (10, 0) v2 = (20, 13) v3 = (60, 28) v4 = (32, 40) = q3 v5 = (32, 19) v6 = (0, 19) p = (21, 21) q2 q4 v1 = (0, 10) v4 = (22, 22) = q4 v3 = (50, 60) v2 = (60, 0) p = (18, 33) v1 = (0, 10) v5 = (25, 35) = q4 = q5 v4 = (49.999, 60) v3 = (60, 0) p = (35,25) v2 = (40, 18)
Fig. 1. Three examples of a test point and a polygon with subdivision dimension 20 20.
Table 1
Process of CBCA for test point inFig. 1.
Example The cell containing p The edges crossing the cell The closest point Test result
Fig. 1a Cell(2, 2) v2v3, v3v4, v4v5 q2 In
Fig. 1b Cell(2, 1) v4v1 q4= v4 In
Fig. 1c Cell(2, 2) v4v5, v5v1 q4= q5= v5 In or Out
Note: the position of a cell is represented by Cell(row, column), and the edges are directed.
S. Yang et al. / Computers & Geosciences 36 (2010) 205–213 206
In this paper, we define a quasi-closest point to represent the closest point in a certain direction, and then present the criteria for obtaining an inclusion property from the quasi-closest point without using the convexity property of a vertex when its adjacent edges are nearly collinear. In Section 3, the definition and criteria are given in a mathematical way so that the orientation method can be applied to the point-in-polygon tests against possibly concave polygons. Based on the uniform subdivision method, Section 4 presents the algorithm for obtaining the quasi-closest point. Especially, if the point obtained by the algorithm is within the neighborhood of a test point for a user-defined tolerance, it is the closest point in any direction. By the minimum distance, the algorithm for an inclusion query does deal with the singular case that a test point is on an edge. Section 5 illustrates the preprocessing steps of point-in-polygon tests for a large number of test points, where the criteria are employed to obtain the inclusion property of an empty cell. The experimental tests as well as the case study in GIS are demonstrated in Section 6. These results exhibit the characteristics of the new method, for example,
it works as efficiently as other similar algorithms do; it handles the singular case that a point is on an edge; and it can be applied to GIS operations such as point-in-polygon
overlay.
3. The criteria for point-in-polygon test based on a quasi-closest point
Following the rule in (Zalik and Kolingerova, 2001), we assume that the polygons are simple ones and oriented, e.g. counter-clockwise. For a simple polygon, the line segment is called edge, and the point where each pair of adjacent segments intersect is called vertex (O’Rourke, 2000).
3.1. A quasi-closest point
Denote by d(p1, p2) the distance between two points p1and p2.
Let p be a point and ei(1rirn) be an edge of a polygon P. Then
the distance from p to eiis defined by
dðp; eiÞ ¼minfdðp; qÞ9qA eig:
If qAeiand d(p, q)= d(p, ei), q is called the closest point from p to ei
and denoted by Q(p, ei).
The closest point from p to ei, say q=Q(p, ei), is called a
quasi-closest point from p to P if for the line segment pq,
(i) pq\ek=| for 1rkrn, kai; or
(ii) pq\ek=| for 1rkrn, kai, j, and q=Q(p, ej).
InFig. 2, q1and q4are quasi-closest points for they meet the
condition (i), whereas q5and q6are not; q2is a quasi-closest point
for it meets the condition (ii), whereas q3is not. Note that for a
test point, there may be several quasi-closest points and one of them is the closest point to all the edges. For convenience, the quasi-closest point meeting the condition (i) is denoted by Q(p, ei,
P), and the one meeting the condition (ii) by Q(p, ei, ej, P), where ei
and ejare adjacent edges (since P is a simple polygon) and the
order of eiand ejagrees with the direction of P.
3.2. Left/right side
For a triangle nv0v1v2, its signed area can be computed as
Aðnv0v1v2Þ ¼ ððx1x0Þðy2y1Þ ðy1y0Þðx2x1ÞÞ=2;
where vi= (xi, yi) for i= 0,1,2. The signed area possesses the
following properties:
(i) If A(nv0v1v2)40, then A(nv1v2v0)40.
(ii) If A(nv0v1v2)40, then A(nv1v0v2)o0.
(iii) If A(nv0v1v2)40, then A(nv0v1p)40, where pAv1v2\{v1}.
For a directed edge e0=v0v1and a point p, the point is on the left
side of the edge if A(nv0v1p) 40, while the point is on the right
side of the edge if A(nv0v1p)o0. For e0 and its adjacent edge
e1=v1v2, e1is on the left side of e0if A(nv0v1v2)40, while e1is on
the right side of e0if A(nv0v1v2)o0.
Denoted by +v0v1v2 the smaller angle formed by v0v1 and
v1v2. Obviously, 0o+v0v1v2r
p
. For v1= Q(p, e0, e1, P), therelationship between +v0v1v2 and the orientation of p with
respect to e0and e1can be stated as follows:
Theorem 1. Let v1=Q(p, e0, e1, P) and pav1 (see Fig. 3). If
+v0v1v24
p
/2, then p is on the same side of e0and e1.Proof. We wish to show that v0and v2are on different sides of
v1p. For v1= Q(p, e0, e1, P) and pav1, we have +v0v1pZ
p
/2 and+pv1v2Z
p
/2. Suppose that +v0v1p=p
. It follows that +v0v1p= +v0v1v2+ +pv1v24p
, contrary to the hypothesis. Therefore,p
/2r+v0v1pop
; by a similar proof,p
/2r+pv1v2op
. If v0and v2are on the same sides of v1p, it follows that +v0v1v2o
p
/2,which contradicts the statement that +v0v1v24
p
/2. So there aretwo cases:
(1) If A(nv1pv0)4 0 and A(nv1pv2)o0, it follows that
A(nv0v1p)40 due to the property (i), and A(nv1v2p)40 due to
the property (ii). This means that p is on the left side of both e0
and e1.
(2) If A(nv1pv0)o0 and A(nv1pv2)40, it follows that
A(nv0v1p)o0 and A(nv1v2p)o0. This means that p is on the
right side of both e0and e1. &
3.3. Point-in-polygon test based on a quasi-closest point
Theorem 2. Let p be a point, P be a simple polygon, and q be a quasi-closest point from p to P. Assume that P is oriented counterclockwise and paq. Then p is inside P if one of the following conditions holds:
(i) p is on the left side of eifor q= Q(p, ei, P).
(ii) p is on the left side of both eiand ejfor q= Q(p, ei, ej, P).
(iii) ejis on the right side of eifor q= Q(p, ei, ej, P).
p q1 q2 q3 q4 q6 q5
Fig. 2. Closest points from a point to edges of a polygon. Among them, q1, q2, and
ARTICLE IN PRESS
Proof. Since paq, we can choose a point p0Apq\{p, q} such that p0
is close enough to q (seeFig. 4). Indeed, by the definition of quasi-closest point, pp0\e
k=| (1rkrn). If p0is inside P, then p is inside
P due to the property of Jordan closed curve.
(i) From the property (iii) of the signed area, since q= Q(p, ei, P)
and p is on the left side of ei, p0is on the left side of ei. As a
result, p0 is inside P by noting that P is oriented
counter-clockwise and p0is close enough to q.
For q=Q(p, ei, ej, P) and qap, ei\ej= {q}. Let C be the circle
centered at q with the radius d(p0, q), {p
i}= C\eiand {pj}= C\ej. Then
C is composed of two circular arcs whose endpoints are piand pj,
that is, C= L(pi, pj)[R(pi, pj), where L(pi, pj) is the circular arc
connecting ei from its left side and R(pi, pj) is the other one
connecting eifrom its right side. Since q= Q(p, ei, ej, P) and qap, we
have p0e{p
i, pj}. If p0AL(pi, pj), then p is inside P by noting that P is oriented counterclockwise and the direction from pito q agrees
with that of P. We now prove that p0
AL(pi, pj).
(ii) For p is on the left side of ei, A(npiqp)40. Suppose p0AR(pi, pj). It follows that pjis inside the triangle formed by eiand p.
Using the properties of the triangle signed area, we obtain A(npjqp)40, A(nqpjp)o0. This means that p is on the right
side of qpj, which contradicts the statement that p is on the
left side of ej. Hence it holds that p0AL(pi, pj).
(iii) Since ejis on the right side of eiand ei\ej= {q}, there exists a
triangle formed by ei and ej. Suppose p0AR(pi, pj). By hypothesis, p0 is inside the triangle so that the vertex q is
not the closest point from p0to e
iand ej. This contradicts the
statement that q is the closest point from p0to e
iand ejdue
to q= Q(p, ei, ej, P). So it holds that p0AL(pi, pj). & A similar proof holds for the following theorem:
Theorem 3. Let p be a point, P be a simple polygon, and q be a quasi-closest point from p to P. Assume that P is oriented
counterclockwise and paq. Then p is outside P if one of the following conditions holds:
(i) p is on the right side of eifor q=Q(p, ei, P).
(ii) p is on the right side of both eiand ejfor q= Q(p, ei, ej, P).
(iii) ejis on the left side of eifor q= Q(p, ei, ej, P).
To use the two theorems, we find first a quasi-closest point. Then, the condition (i) is checked if the quasi-closest point does not coincide with any polygon vertex; otherwise, the conditions (ii) or (iii) is checked sequentially. Due to Theorem 1, the condition (iii) is checked only when the adjacent edges are far from co-linearity. Therefore, the theorems give the criteria for point-in-polygon test, which does not need a tolerance to determine the convexity property of a vertex. Furthermore, the advantage of the criteria is that only the sign of the area is required.
It is worth noting that the two theorems can be applied to the region enclosed by a finite number of simple polygons no matter whether there are holes or not. In such case, the polygons should be oriented so that the interior region always lies on the same side of its edges (seeZalik and Kolingerova, 2001). The proof of Theorem 2 also indicates that for a test point, an identical query result can be obtained from different quasi-closest points. In the following, the process of obtaining the result of point-in-polygon test from a quasi-closest point is simply called an inclusion query, and we will focus on finding a quasi-closest point neighboring to a test point.
4. Finding a quasi-closest point
Let P be a polygon and R(P) be the bounding rectangle of P. Then R(P) can be subdivided into a grid of sub-rectangles with uniform size. In the following, the sub rectangle is called the cell of subdivision.
For a point pAR(P), we use Cell(p) to denote the cell containing p.Fig. 5illustrates Cell(p), its eight neighboring cells, and the cells overlapped by the
d
-neighborhood of p (which will be simplyp v0 v1 = Q (p, e0, e1, P) e0 e1 v2 p v0 v1 = Q (p, e0, e1, P) v2 e0 e1
Fig. 3. Illustration for the proof of Theorem 1 (p is on same side of e0and e1for +v0v1v24p/2).
p p’ q pi pj ei ej p p’ q ei
Fig. 4. Illustration for the proof of Theorem 2. (a) q = Q(p, ei, P); (b) q = Q(p, ei, ej, P).
S. Yang et al. / Computers & Geosciences 36 (2010) 205–213 208
called the
d
-neighborhood cells). In particular, if a cell contains a subset of points of an edge, we say that the cell is crossed by the edge. For convenience, we use Cells(e) to denote the cells crossed by the edge (or line segment) e, and Edges(C) to denote the edges crossing the cell C. If a cell is not crossed by any edge, we say that the cell is empty.Based on the subdivision, a quasi-closest point can be obtained from the following result:
Theorem 4. Let pAR(P) and q= Q(p, ei). Then q is a quasi-closest
point if q is the closest point from p to the edges crossing the cells Cells(pq), that is,
d(p, q)= min{d(p, q0)9q0Ae, eAEdges(C), CACells(pq)}.
Proof. This result is trivial if p = q. For paq, suppose that there exists an edge ej(jai) such that pq\ej= {q
n
}. It follows that Cell(qn
)ACells(pq) due to qn
Apq. Noting the condition, we have q= Q(p, ej).
(a) If qaqn
, it follows that d(p, qn
)od(p, q), contrary to the condition. Hence we obtain pq\ek=| (1rkrn, kai).
(b) If q =qn
, it follows that q is the vertex connecting eiand ejdue
to that P is a simple polygon. As a result, pq\ek=| (1rkrn,
kai, j).
By the definition of a quasi-closest point, we obtain the result. &
From Theorem 4, an algorithm is developed for obtaining a quasi-closest point; from Theorem 2 and 3, the algorithm of an inclusion query is presented in the following as well:
Algorithm: Finding aquasi-closest point.
Input: a point p, a set of cells initcells (composed by one or several cells neighboring to p)
Output: a quasi-closest point q
0: curcells’initcells
1: Find the closet point of p to the edges crossing curcells, say q 2: Set a flag for each of curcells to indicate that it has been used 3: If each of Cells(pq) has been used, then return q
4: curcells’Cells(pq), go back to step 1
Algorithm: Perform an inclusion query (for a user-defined tolerance
d
).Input: a point p, a quasi-closest point q Output: IN, OUT or ON
1: If d(p, q)r
d
, return ON 2: If q= Q(p, ei, P) then3: If p is on the left side of ei, return IN
4: Else if q=Q(p, ei, ej, P) then
5: If p is on the left side of both eiand ej, return IN
6: If p is on the right side of both eiand ej, return OUT
7: If ejis on the right side of ei, return IN
8: End If 9: return OUT
Now, we make some remarks on the algorithm for finding a quasi-closest point.
(1) The initial cells of the algorithm are either one of the non-empty neighboring cells or the
d
-neighborhood cells for a user-defined toleranced
. The former is used in the preproces-sing stage to determine the inclusion property of an empty cell, and the latter is used in the processing stage to determine the inclusion property of a test point. Note that the latter is used only when some of thed
-neighborhood cells are non-empty. As a result, for a test point p and the quasi-closest point q, q is the closest point of p to all the edges if d(p, q)rd
. From the distance d(p, q), the algorithm of an inclusion query can resolve the singularity that a test point is on an edge. (2) The algorithm really returns a quasi-closest point and itscorresponding edge(s), which will be used for the inclusion query.
(3) The loop of the algorithm would terminate in a finite number of steps because the initial cells are composed of one or several of the neighboring cells.
(4) The average-case time complexity depends on the average number of the edges crossing a non-empty cell. In the real applications of GIS, the average number is almost a constant independent of the number of the polygon edges. Thus, the time complexity can be regarded as a constant.
Combining the two algorithms, we obtain a new method for point-in-polygon test based on a quasi-closest point. For con-venience, the method is named QCPM (abbreviated for Quasi-Closest Point Method). QCPM requires two parameters, that is, a test point and initial cells; its result will indicate the location of the point such as on an edge, inside or outside a polygon. In particular, whether a point is on an edge is determined by the minimum distance.
5. Point-in-polygon tests for a large number of points
To accelerate the point-in-polygon tests for a large number of points, we preprocess a polygon using the method of Zalik and Kolingerova (2001). First, the bounding rectangle of a polygon is C4 C3 C2
C5 Cell (p) C1
C6 C7 C8 p
Fig. 5. Illustration for a test point p, the cell Cell(p) containing p, and eight neighboring cells of Cell(p). Among these cells, cells overlapped byd-neighborhood of p are indicated in grey color.
ARTICLE IN PRESS
subdivided into uniform-sized cells. Let n be the number of the polygon edges, W and H be the width and height of the bounding rectangle, respectively. Then the dimension of a cell is determined by Sizex¼ W NoOfCellx ; Sizey¼ H NoOfCelly ; where NoOfCellx¼2 Wpffiffiffin H ; NoOfCelly¼2 Hpffiffiffin W :
Next, each edge of the polygon is stored in the cells crossed by its corresponding line segment. The crossed cells can be enumerated by a variant of the traversing algorithm (Bresenham, 1965;Fujimoto and Iwata, 1983;Cleary and Wyvill, 1988;Zalik et al., 1997). In addition, we add a flag variable for each cell to indicate whether it has been used.
Finally, the empty cells are classified as inside or outside a polygon. Here, QCPM is used for the classifications when necessary. As shown in the following algorithm, when an empty cell does not neighbor to any cell already classified or out of the bounding rectangle, QCPM is employed with the center point of the empty cell as its test point and one of the non-empty neighboring cells as its initial cells.
Algorithm: Classify the empty cells for the bounding rectangle R(P).
For each cell C of R(P) If C is empty
If one of the eight neighboring cells of C is out of R(P) Label C with OUT
Else If one of the eight neighboring cells of C has been labeled
Label C with the inclusion property of the labeled cell Else //some of the neighboring cells of C are non-empty Find the quasi-closest point for the center point of C with one of non-empty neighboring cell as the initial cells
Label C with the result obtained by an inclusion query End If
End If End For
Based on the preprocessing result, the inclusion properties of a large number of test points can be determined efficiently. For a test point p and a user-defined tolerance
d
, if all thed
-neighborhood cells are empty, the property of p is obtained directly from that of Cell(p); otherwise, the property is determined by QCPM with thed
-neighborhood cells as the initial cells. Therefore, the two classifications, one for an empty cell and the other for a test point, are united in this method.6. Experiments
The algorithms proposed in this paper have been implemented in Visual C++ language, and the experiments were performed on a PC with 512 M RAM and a Pentium CPU (2.93 GHz).
6.1. Experimental tests
We perform the point-in-polygon tests with 1000 1000 points evenly distributed in the bounding rectangle of polygon(s). As shown inFig. 6, an artificial polygon is used for the first set of experiments, which give counterexamples to the method proposed by (Zalik and
Kolingerova, 2001). The counterexamples come from the two areas emphasized inFig. 6c, where the closest point found by the algorithm is either outside the cell containing the test point or coincident with a polygon vertex.Fig. 6d shows the results from the new method with a larger user-defined tolerance, and among the 1,000,000 test points, 133,119 points are lying on an edge. The comparative experiment shows that the new method appears to offer stability over a similar method and moreover consistently deals with the case of a test point on an edge.
In the second set of experiments, we investigate the performance of the new method.Fig. 7shows the polygons presented by (Zalik and Kolingerova, 2001). The running time of the point-in-polygon tests performed on these polygons is listed inTable 2. Besides,Table 2also lists the two factors that affect the performance of the new algorithm, where
l
is the ratio of the non-empty cells to all the cells, andk
is the average number of the edges stored in a non-empty cell. The same experiments are also performed on the artificial polygons (seeFig. 8andTable 3), which are randomly generated in the method described byLi et al. (2007). From the data inTables 2 and 3, we observe that for the edge number n, as n goes larger inTable 2,
l
becomes smaller and the processing time lessens; as n goes larger inTable 3,k
becomes larger and the running time increases. Obviously, the data inTable 2simulate the real applications whereas the data inTable 3hardly arise in reality. Anyway, these experiments confirm that the factor
k
can be regarded as a constant independent of n, and on average the preprocessing time is linear with n. In particular, the processing time is a constant independent of n, that is, an inclusion test requires constant time.In the final set of experiments, we compare the running time of different algorithms. Table 4 lists the running time of two algorithms, where Zalik’s algorithm comes from (Zalik and Kolingerova, 2001) and its program is downloaded from the website http://www.iamg.org/index.php? option=com_content& task=view&id=175&Itemid=165. Note that the cell number of rows or columns is limited to 100. Although the new algorithm runs faster, we believe that the difference comes from the different implementations. Nevertheless, the results show that the efficiency of the new algorithm is comparable with that of a similar algorithm.
6.2. Case study
The new method can be applied to point-in-polygon overlay in GIS. The point-in-polygon overlay is a spatial operation in which points from one feature dataset are overlaid on the polygons of another to determine which points are contained within the polygons (seeFig. 9).
The case study uses two shape files. The shape file for polygons includes 23266 edges as well as 61 polygons (two of them are holes), and the shape file for spots includes 1377 points. The aim of the study is to test the capability of the method to identify which region contains a test point in a map. The overall workflow is illustrated as follows:
1. Read the map from a shape file.
2. Form the administrative boundaries into oriented polygons. 3. Preprocess the map by subdividing its bounding rectangle into
grid cells, recording the borderlines into the cells, and identifying the region to which each empty cell belongs. 4. Read the spots from a shape file.
5. Identify the region to which each spot belongs.
At step 2, it is straightforward to represent the administrative regions in oriented polygon because the polygons in shape file dataset are already oriented. However, the polygons of a map are S. Yang et al. / Computers & Geosciences 36 (2010) 205–213
generally connected, so we have to merge the coincident edges such that the two sides of the merged edge belong to different regions. InFig. 9, the merged edges are indicated in blue color, and the edges in red color indicate that one of its sides is out of the
whole country. Especially, two numbers are assigned to each edge for identifying the regions on the left and right sides.
Now, we revise the algorithm for an inclusion query so that the revised algorithm can return a number for identifying the region on the left or right side. For a quasi-closest point such as Q(p, ei, P)
or Q(p, ei, ej, P), the identification number can be obtained directly
from the edge ei. If a quasi-closest point is coincident with a
vertex connecting more than two edges, we have to select two edges for the inclusion query. InFig. 10, for example, we select the two edges adjacent to Region 1 if p is a study spot. For the study spot p and its quasi-closest point q, the selection can be accomplished by evaluating the angles between pq and the connected edges. Thus, we obtain the revised method for identifying the polygon to which a test point belongs.
At step 3, the bounding rectangle of the map is subdivided into a group of grid cells, and each borderline of the map is recorded Fig. 6. Point-in-polygon test results (Red-inside, Green-outside, Black-on edge) (a) an artificial polygon; (b) cells of uniform subdivision; (c) test results from (Zalik and Kolingerova, 2001); (d) test results from new method for a larger user-defined tolerance.
Fig. 7. Polygons and the subdivision from (Zalik and Kolingerova, 2001).
Table 2
Running time on polygons inFig. 7.
Polygon Edge number Cell number l k Running time (s) Preprocess Process a 10 48 0.67 1.375 0.000044 0. 972392 b 149 684 0.42 1.575 0.000414 0. 801133 c 1248 5060 0.31 1.921 0.003410 0.722073 d 28012 112,992 0.12 3.097 0.054073 0.581358
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into the cells crossed by the borderline. Then, the revised method is used to identify the region to which an empty cell belongs, and correspondingly, an identification number of the region is stored in each empty cell.
At step 5, the revised method is also employed to identify the region to which a spot belongs when necessary. The results are visualized in Fig. 9, where the spots in different regions are indicated by different colors. For the case with 1377 spots and 61 regions, the revised method is used 1377 times. Other methods, such as the ray crossing number method or the sum of angles method, can answer the point-in-polygon problem, but cannot directly answer such problems as to which polygon a point belongs. Using these methods, we can identify the polygon by point-in-polygon test for each polygon of the map; however, for the overlay operation, such tests need to be performed 1377 61/2 times on average.
7. Conclusion
This paper has presented a robust solution to the point-in-polygon problem. The solution employs a quasi-closest point that can be locally found by the uniform subdivision cells. Based on the quasi-closest point, an inclusion query obtains its result from the sign of triangle signed area, and avoids using a dubious tolerance to identify the convexity property of a polygon vertex. Although obtaining a quasi-closest point requires a little more time than finding the closest point from an individual cell, the extra time cost is rewarded with the correct answer to the point-in-polygon problem as well as the consistent treatment of the singular case that a test point is on an edge.
In this paper, the two classifications (one in the preprocessing stage and the other in the processing stage) are achieved by the inclusion query based on a quasi-closest point. This simplifies the Fig. 8. Polygons randomly generated.
Table 3
Running time of polygons inFig. 8.
Polygon Edge number Cell number l k Running time (s)
Preprocess Process a 50 224 0.50 1.770 0.000203 0.888035 b 500 2116 0.58 3.423 0.002139 1.157722 c 1001 4224 0.65 4.683 0.005793 1.412953 d 5000 20448 0.68 9.856 0.049777 2.145732 Table 4
Running time of different algorithms (in seconds).
Polygon(s) inFig. 7 Edge number Cells of subdivision k Zalik’s algorithm The new algorithm Preprocess Process Preprocess Process a 10 6 8 1.375 0.000055 2.555309 0.000044 0.972392 b 149 18 38 1.575 0.000645 1.839879 0.000414 0.801133 c 1248 46 100 1.988 0.004844 1.707092 0.002827 0.740552 d 28,012 100 100 9.200 0.042282 7.483888 0.024614 1.205676
S. Yang et al. / Computers & Geosciences 36 (2010) 205–213 212
task of programming and distinguishes from the majority of other approaches.
The present study demonstrates the feasibility of the point-in-polygon test based on a quasi-closest point. In the future, we will apply the method to an inclusion test for the planar region enclosed by the curves such as circular arcs and free-form curves.
Acknowledgements
The research was supported by Chinese 973 Program (2004CB719400), the National Science Foundations of China (60625202, 90715043) and Chinese 863 Program (2007AA-040401). The second author was supported by the project sponsored by a Foundation for the Author of National Excellent Doctoral Dissertation of PR China (200342) and the Fok Ying Tung Education Foundation (111070). In addition, the authors would like to thank the reviewers for their helpful comments.
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Fig. 10. Quasi-closest point connecting more than two edges. Fig. 9. Point-in-polygon overlay on map of Germany.
